# Talk:Closed Form Precision

For the most part the discussion of the Bessel, Gaussian and Rayleigh correction factors is off the deep end. All elementary text books would use (n-1) as the divisor when calculating the standard deviation, but few would name it. The Gaussian correction is small compared to the overall size of the confidence interval with small samples. For measures like group size it is also unnecessary. For small samples the distribution is significantly skewed so you should use tables based on Monte Carlo results which would have the correction factor "built in." By the time you get enough data for group size measurements to be really be normally distributed the Gaussian correction factor is then totally negligible.

Perhaps the correction factors could be buried deeper or on another page. But we can't just wave them away and we shouldn't pull mysterious correction factors out of thin air. This is, after all, the closed form analysis. Yes, you can get the same results via Monte Carlo and use that approach instead, but for those who want to see the pure math these terms are part of it. David (talk) 17:56, 24 May 2015 (EDT)

The fact that most of the measures are really skewed distributions deserves more serious discussion.

Agreed. Higher moments are noted in a few of the spreadsheets, like Media:Sigma1RangeStatistics.xls, but a discourse would be a worthwhile addition. David (talk) 17:56, 24 May 2015 (EDT)

I must admit a bit of curiosity here. I don't know if the Student's T test has the Gaussian correction built into the tables or not. For small samples (2-5 degrees of freedom) the confidence interval is so large that it wouldn't make a lot of difference, and when doing various error combining to get an "effective" number for the degrees of freedom then having factor built in might mess things up. I looked and can't find the answer readily available. I'll try to remember to ask the question on the math stats forum.

The paragraph under heading "Estimating σ" misses the salient points entirely. First there are so many different uses of $$\sigma$$ that it is confusing when hopping around on wiki. $$\sigma_{RSD}$$ should be used consistently for the radial standard deviation of the population of the Rayleigh distribution. Second $$s_{RSD}$$ can't be calculated directly from the experimental data like s can for the normal distribution. You have to calculate $$s_r$$ from the data, then use a theoretical proportionality constant to get $$s_{RSD}$$.

As noted at the top of the page this use of σ is not confusing because it turns out to be the same σ used in the parameterization of the normal distribution used as the model, which is the same as the standard deviation. David (talk) 17:56, 24 May 2015 (EDT)

The discussion about the unbiased Rayleigh estimator is just wrong. I'll admit another limit on my knowledge here. You could calculate $$s_{RSD}$$ from either the experimental $$\bar{r}$$ or $$s_r$$. It seems some combination of the two would give the "best" result. I don't have a clue how to do that error combination. I do know that in general the measurement of $$\bar{r}$$ would have a greater relative precision (as % error) than $$s_r$$. In fact the ratio of the two estimates could be used as a test for outliers. The standard deviation is more sensitive to an outlier than the mean.

I don't understand what's wrong: the math is right there in Closed Form Precision#Variance_Estimates and Closed Form Precision#Rayleigh_Estimates. Expand the equations and they are identical. David (talk) 17:56, 24 May 2015 (EDT)

In the section on Mean Radius the use of the phrase Mean Diameter seems out of whack. Using a "diameter" as a radius is just confusing for no good reason. Seems like you should just use 2$$\bar{r}$$.

Mean Diameter has come up as an attractive standard, since it corresponds to the 96% CEP which is closer to the measures people are used to than 50% CEP or the mean radius, and rolls off the tongue more easily than "two mean radii" or any other variation I've heard of. David (talk) 17:56, 24 May 2015 (EDT)

A number of things are stuffed on this wiki page because they fit some erroneous notion of "Closed form". For example the discussion of sighter shots. That should be a a separate wiki page. The whole thing about the errors for sighter shots also depends on the assumptions for the dispersion for which there are 4 general cases.

Yes, we should probably break that onto a separate page. David (talk) 17:56, 24 May 2015 (EDT)

Herb (talk) 16:36, 24 May 2015 (EDT)